$ A = \left[\begin{array}{rr}1 & -1 \\ 0 & 1\end{array}\right]$ $ C = \left[\begin{array}{rr}0 & -1 \\ 1 & 0\end{array}\right]$ Is $ A C$ defined?
In order for multiplication of two matrices to be defined, the two inner dimensions must be equal. If the two matrices have dimensions $( m \times  n)$ and $( p \times q)$ , then $ n$ (number of columns in the first matrix) must equal $ p$ (number of rows in the second matrix) for their product to be defined. How many columns does the first matrix, $ A$ , have? How many rows does the second matrix, $ C$ , have? Since $ A$ has the same number of columns (2) as $ C$ has rows (2), $ A C$ is defined.